MODELS AND PREDICTION ALGORITHMS OF FRACTURE OF

STRUCTURAL ELEMENTS FOR LOW- AND HIGH-CYCLE LOADING

BASED ON FEM

S.A. Kapustin*, V.A. Gorokhov, Yu.A. Churilov

Research Institute for Mechanics of Lobachevsky State University of Nizhni Novgorod,

23 Prospekt Gagarina (Gagarin Avenue) BLDG 6, Nizhny Novgorod, 603950, Russia

*e-mail: sergei.kapustin@mail.ru

Abstract. Mathematical models, methods, algorithms and developed based on them the

results of numerical simulation based on FEM, approaches mechanics of the damaged

medium, processes of deformation and fracture (from the generation stage of microdefects to

the propagation of the main cracks) of structural elements for low- and high-cycle loading.

1. Introduction

Modern approaches to the analysis of the structural strength of the most widely used methods

of linear and nonlinear fracture mechanics, studying the final stage of destruction -

propagation trunk cracks. Much less attention is given to the study of the initial period of the

destruction associated with the accumulation of damage. However, the crack originates and

develops in the volume of material prepared for destruction within the first, initial stage.

Therefore, the modeling of real processes of destruction of structures should ensure that the

description of these processes in both the early and final stages.

In the present work we study the deformation and damage accumulation in the material

of constructions is carried out in the framework of the ratios of the mechanics of the damaged

medium using proposed by authors the composite hierarchical model of the damaged material

[1], which allows to investigate the behavior of structures with consideration of the

peculiarities of the processes of destruction at the initial and final stages. The model is based

on the ability to represent complex process of propagation of the interrelated effects of

deformation and fracture in a series of formally independent elementary acts described by

respective partial models of plasticity and damage accumulation. Calculation of mutual

influence such basic acts is at the top level in the general model of the damaged material.

However, the description of the interaction of different types of damage and their effect on the

deformation process is based on invariant with respect to the nature of these injuries scalar

measures of damage ω .

2. The model of the damaged material and methodic the numerical solution of problems

of fatigue strength of structures

The overall proportions of the model of the damaged material, establish the link between

changes in the reduced stresses and strains at the elementary step of external influences

changes [1, 2]. They have the form of equations of elasticity with additional members, due to

the effects of temperature, irreversible deformation and degradation of material properties

associated with the accumulation of damage. In the framework of a general model of the

damaged material implemented private models thermoplasticity with combined hardening and

Materials Physics and Mechanics 23 (2015) 79-82 Received: March 27, 2015

© 2015, Institute of Problems of Mechanical Engineering

damage accumulation under low-cycle thermo loadings [1], and the model of fatigue damage

accumulation during under high-cycle thermo loadings taking into account the dependence of

limit cycles from the level of the operating temperature of the asymmetry parameter cycle

implemented in a loop of the stress state, as well as measures of the accumulated damage [3].

To describe the development of damage in the material under cyclic loading is

introduced function ψ, representing the normalized counterpart of hazardous energy, and

scalar measure of damage ω used to describe the influence of current damage on the

characteristics of the deformation process on the basis of the hypothesis about the existence of

two phases of damage accumulation [1, 3, 4]. Within the first-hand phase is the emergence

scattered by the volume of material damage in the vie de micropores and microcracks that

does not lead to a significant impact of these injuries on the physico-mechanical

characteristics of the material. In relation to the above model of the damaged material for this

phase can be considered to change the measure of damage Δω 0 . The second phase is

characterized by further development of co-interaction of defects occurring and is

accompanied by the growing influence of the damage onto the physico-mechanical

characteristics of the material and destabilization of the process of cyclic deformation under

cyclic loading. The end of phase conforms the appearance in the material of macroscopic

cracks.

Simulation of fatigue crack propagation in the construction carried out by the

"shutdown" nodes when reaching the critical values of the measures of damage. The process

of propagation of the crack is considered as a sequential "shutdown" of neighboring nodes

during loading process [2], without modifying the original scheme of discretization and the

initial topology of the investigated structures.

Numerical solution of boundary value problems of deformation and fracture of

structural elements is based on the FEM using the generic isoparametric models FE, with high

efficiency in the analysis of both massive and thin-walled fragments [1].

3. Algorithms of the prediction of fracture under cyclic loading

The necessity of creation of such algorithms is due to the fact that direct application scheme

described above to solve the problems of modeling the behavior of structures under cyclic

loading, by successive integration of the equations for a large number of cycles, it is very

difficult, both because of the great complexity of the calculations, and the possibility of

accumulation of numerical errors. To overcome these difficulties, the proposed prediction

algorithms process of origin and propagation of fatigue cracks under cyclic loading based on

the numerical simulation of these processes in the framework of the ratios of the mechanics of

the damaged medium. These algorithms [4, 5] can significantly reduce the overall complexity

of the numerical solution of problems of fatigue strength of structures.

On the foundation of the algorithms the possibility of extrapolating on the load cycles of

the parameters characterizing the current elastic-plastic condition and damage of material in

the construction nodes is based, taking into account the hypothesis of multi-stage nature of the

development of damage in the material during deformation. The proposed algorithms consist

of two levels of extrapolation: linear – at the first stage of damage accumulation; nonlinear in

the second stage of damage accumulation.

Under low-cycle loading within the first stage can occur stabilization process of cyclic

deformation occurring in the stabilization of the amplitude values of stress, strain, plastic

strain. On the section of the process from the beginning of stabilization until the end of the

first stage of damage accumulation, corresponding to the value ψ ψа (ψa – amplitude value

function damage, specifying the moment of completion of the first phase of the development

of fatigue damage in which the cumulative damage does not affect the mechanical properties

of the material), the process parameters in elastic-plastic deformation is practically constant

80 S.A. Kapustin, V.A. Gorokhov, Yu.A. Churilov

(amplitude values of stresses, full and plastic deformation), and the length of the trajectory of

plastic deformation pk (Odkwist parameter) and damage function ψ , change in law, is close

to linear. Therefore, at this stage loading can, with reasonable certainty, to predict values of

the above parameters by linear extrapolation on EXT1 number of cycles forward, eliminating

the corresponding part time-consuming process step-by-step integration of the equations of

the original problem [4].

In the case of high-cycle loading, by results of numerical simulation of the deformation

process of construction for the first loading cycle and calculating the increment of function

damage ψ , extrapolation on EXT1 loading cycles is, which corresponds to the completion of

the first phase of damage accumulation in the most loaded physical node structure [5].

In the second stage of destruction when ψ ψа , the collective interaction of the

developing micropores and microcracks is in the material, leading to a significant influence of

the accumulated damage on macroscopic material characteristics (modulus of elasticity, sound

velocity, and so on), growth measures of damage ω , culminating in the formation of

microdefects (ω 1.0 ).

At this stage of damage accumulation the extrapolation of the parameters of the damage

of the material ψ and ω on the number of cycles EXT2 is, given by set changing of the

measure of damage maxω in the most loaded node of construction, and subsequent

refinement of the equilibrium structure [5]. The expression to determine the number of cycles

extrapolation 2EXT can be obtained from the kinetic equations of fatigue damage

accumulation [1, 5]. Extrapolation of the second level can be performed repeatedly until a

measure of the damage in some physical node of construction ω reaches a critical value that

would indicate the emergence of macroscopic cracks.

The proposed algorithms for the prediction of low- and high-cycle of destruction of

structures implemented in the framework of the established and developed in the Institute of

mechanics of the Nizhny Novgorod University computing complex UPAKS [6].

4. The results of numerical simulation of fatigue failure of structural elements

To assess the performance and computational efficiency of the proposed algorithms for

predicting fracture and created on their basis of software tools for the numerical investigation

of fatigue strength of structures in this section, we present some results of calculations of

fracture construction elements for low- and high-cycle loading.

Numerical simulation of low-cycle fracture of the cylindrical sample. An example

of numerical simulation of low-cycle deformation and fracture of cylindrical sample, made of

stainless steel 12Kh18N10T with working part of the recess [4]. The sample is uniformly

heated by volume to a temperature of T = 350 °С and is in conditions of cyclic loading by

applied axial displacement on the end varying according to the law of the symmetric cycle.

It was decided two versions of the task. In the first variant numerical study of

deformation and damage accumulation was carried out until the crack formation without the

use of extrapolation procedures. It was found that the measure of damage in the most loaded

point has reached the limit ω 0.99 of 1040 cycle. When the decision of the second variant

of the problem using extrapolation procedures (20 cycle of loading was produced

extrapolation 750 cycles) the predicted number of cycles to failure was 1067.

Thus, the relative error in the determination of the maximum number of cycles using

extrapolation procedures for the considered tasks does not exceed 2.7 %. Using extrapolation

procedures have helped to reduce the complexity of the computational process by 70 %.

Numerical simulation of high-cycle fatigue of the experimental sample. In this task

was carried out numerical study of high-cycle fatigue of the experimental sample, made of

steel VJ-159. The results of calculations and their comparison with experimental high-cycle

81Models and prediction algorithms of fracture of structural elements ...

fatigue curves for the symmetric and asymmetric cycles obtained for the considered material

at a temperature of Т = 850 °С given in [3].

For symmetric loading cycle, corresponding variant of the load at which the

experimental sample is destroyed by 1000000 cycle of loading, the use of the developed

algorithm extrapolation [5] has allowed to describe the process of damage accumulation in the

second stage duration 6000 cycles by twenty extrapolations (when setting maxω 0.05 ).

Table 1 shows the parameters of the extrapolation of the damage accumulation process until

the formation of microdefects in the most loaded point of the sample.

Table 1. The sequence of extrapolations of the second level until formation of cracks.

Number of

extrapolation

N of the

cycle EXT2

Number of

extrapolation

N of the

cycle EXT2

1 990695 2235 11 996663 27

2 992931 1488 12 996691 18

3 994420 837 13 996710 12

4 995258 514 14 996723 8

5 995773 330 15 996732 5

6 996104 215 16 996738 3

7 996320 142 17 996742 2

8 996463 94 18 996745 1

9 996558 62 19 996747 1

10 996621 41 20 996749 1

The data presented show efficiency and high efficiency of the developed algorithms to

predict processes high-cycle destruction of structural elements.

5. Conclusion

Developed models, algorithms, and software tools for the numerical investigation of the

processes of fatigue failure of structural elements within the approaches of mechanics

damaged medium on the basis of FEM with quasi-static thermo loading. Proven efficiency

and effectiveness of the established tools for forecasting processes low- and high-cycle

fatigue of structural elements is.

Acknowledgements. The work was partially funded through the core part of the public task of

the Ministry of education and science (project No. 2014/134 2226) and RFBR (project No.

14-08-31084-mol_a).

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82 S.A. Kapustin, V.A. Gorokhov, Yu.A. Churilov